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Hypersonic Triangle Array Notation
The Hypersonic Triangle Array Notation is one final (real final) extension to this function. Tr|0|(n) = n Tr|1|(n) = Trω(n) For a>1, Tr|a|(n) = Trω(Trω(...Trω(n))..)), iterated Trω(n) times. Rules Edit -all 0's are ignored -for any 1's in the array, solve the array without them, then at the end result apply a Trω(n) function for every 1 in the array. Tr|a,b|(n) = Tr|a|(Tr|a|(Tr|a|(...Tr|a|(n))..)), iterated Tr|b|(n) times. Tr|a,b,c|(n) = Tr|a,b|(Tr|a,b|(...Tr|a,b|(n))..)), iterated Tr|c,c|(n) times. Tr|a,b,c,....,x,y|(n) = Tr|a,b,c,....,x|(Tr|a,b,c,....,x|(...Tr|a,b,c,....,x|(n))..)), repeated Tr|y,y,y,....,y,y|(n) (the amount of y's in the array is equal to the amount of elements originally in the array, -1. Tr|#|(n) = Tr|n,n,n,n,....,n,n,n|(n), with Tr|n+1|(n) n's in the array. Tr|#1|(n) = Tr|#|(Tr|#|(...Tr|#|(n))..)), iterated Tr|#|(n) times. Tr|#a|(n) = Tr|#a-1|(Tr|#a-1|(...Tr|#a-1|(n))..)), iterated Tr|#a-1|(n) times. Tr|##|(n) = Tr|#n|(n) Tr|##1|(n) = Tr|##|(Tr|##|(...Tr|##|(n))..)), iterated Tr|##|(n) times. Tr|##a|(n) = Tr|##a-1|(Tr|##a-1|(...Tr|##a-1|(n))..)), iterated Tr|##a-1|(n) times. => The same rules as with the original #. Then Tr|@|(n) = Tr|###....###|(n), with n #'s in the array. We have the same EXACT rules with the triangularions in the array as with the normal ones. The last thing in the previous notation was Trω(n). Eventually, we would reach: Tr|ω|(n), which is defined as Tr|^4|(n). Same thing as before. Here is where the new stuff appears: Tr|-A-|(n) = Tr|ω|(Tr|ω|(....Tr|ω|(n))..)), repeated Tr|ω|(n) times. Tr|-A+1-|(n) = Tr|-A-|(Tr|-A-|(...Tr|-A-|(n))..)), repeated Tr|-A-|(n) times. Tr|-A+2-|(n) = Tr|-A+1-|(Tr|-A+1-|(...Tr|-A+1-|(n))..)), repeated Tr|-A+1-|(n) times. Tr|-A+3-|(n) = Tr|-A+2-|(Tr|-A+2-|(...Tr|-A+2-|(n))..)), repeated Tr|-A+2-|(n) times. I think you get the iteration style here. Tr|-A2-|(n) = Tr|-A+n-|(Tr|-A+n-|(...Tr|-A+n-|(n))..)), repeated Tr|-A+n-|(n) times. Tr|-A2+1-|(n) = Tr|-A2-|(Tr|-A2-|(...Tr|-A2-|(n))..)), repeated Tr|-A2-|(n) times. I think you get where this is going. Tr|-A3-|(n) = Tr|-A2+n-|(Tr|-A2+n-|(....Tr|-A2+n-|(n)))..))), repeated Tr|-A2+n-|(n) times. Tr|-A*m-|(n) = Tr|-(A*m-1)+n-|(Tr|-(A*m-1)+n-|(...Tr|-(A*m-1)+n-|(n))..)), repeated Tr|-(A*m-1)+n-|(n) times. Tr|-A*A-|(n) = Tr|-A*n-|(n) Tr|-A^A-|(n) = Tr|-A^n-|(n) I think you get how this works because it's similar to FGH. The rules with "A" are the same with omega in the FGH. (except here, it is iterated more times than just "n"). (I hope I am explaining this clear enough so that you understand it. With this new kind of letter notation, we can define recursive ordinals with B,C,D,.... Eg: \(A\) =ω \(B\) = \(\varepsilon_0\) C = \(\zeta\) After reaching Z, we can continue with Z1 to define more recursive ordinals. X however, is a different thing not to be confused with an ordinal. The ordinals in this notation use all the letters from A-Z, except "X". Tr|-X-|(n) = The ZTr|-Zn-|(n)th recursive ordinal (put inside the array, of course, I just didn't put it because subscripts get messy lol) Tr|-X+1-|(n) = The ZTr|-X-|(n)th recursive ordinal Tr|-X+a-|(n) = The ZTr|-(X+a)-1-|(n)th recursive ordinal Tr|-X2-|(n) = The ZTr|-X+n-|(n)th recursive ordinal Define a Za = An infinite subsript nesting of Za-1. This works with every ordinal, for example: C = BB...B, which corresponds to \(\zeta\) = an infinite subscript nesting of Epsilons. In the array, the infinite powertower will be n high. The same rules for other ordinals based off A,B,C, etc as with X CLEARER DEFINITION In this array notation, if we have a LIMIT ORDINAL, then we diagonalize that ordinal (with n, of course.) The Wainer Hierarchy is the normal one used for this notation, however, the Veblen Hierarchy can be attributed to it by specification. Category:Notations Category:Functions Category:MachineGun's Functions